Abstract

The problem posed in this paper is that of restoring a Poisson-point-process intensity that has been degraded by a band-limiting filter followed by a truncation of the signal. The approach is to derive a maximum-likelihood estimate from the count data of the degraded point process. The expectation-maximization algorithm is used to realize this estimate, while the derivation of this algorithm is an extension to previous developments by Shepp and Vardi [
IEEE Trans. Med. Imaging MI-2,
113 (
1982)], Snyder et al. [
IEEE Trans. Nucl. Sci. NS-28,
3575 (
1981)], and others used for positron-emission tomography. We also extend our own work reported earlier by considering the truncated signal, which is analogous to practical situations in both two- and three-dimensional microscopy in which the image of the specimen has been truncated. Computer simulations with one-dimensional and two-dimensional signals demonstrate such a reconstruction with reasonable success. The plausibility of doing such a reconstruction is explained in that for the noiseless case the transformation characterizing the degradation is invertible.

Figures (9)

(a) Signal to be restored. This signal represents the point-process intensity λ. Although this is a continuous signal according to our model, it is represented in the simulation by a discrete array with a sample spacing of 32 nm. The x axis on this plot shows the indices of these samples such that the total spatial width of the 64-element array is 2.048 μm. (b) Degraded signal without truncation. This is the noiseless one-dimensional analogy of a diffraction-limited noncoherent image from a system with a numerical aperture of 1.25 and a wavelength of 525 nm. (c) Truncated signal. This signal represents the analogy of collecting an image of an object that is larger than the viewing window.

(a) Restored signal from the degraded signal of Fig. 1(b) following 1000 iterations of the algorithm in Ref. 1. (b) Restored signal from the degraded signal of Fig. 1(c) following 1000 iterations of the algorithm in Ref. 1.

(a) Initial guess
λ^(0)(x) used in the reconstruction from Fig. 1(c) with the extended algorithm of Section 2. Reconstructions at 100, 1000, and 10,000 iterations are shown in (b), (c), and (d), respecitively.

(a) Different initial guess from that shown in Fig. 3(a). Corresponding reconstructions using the extended algorithm of Section 2 at 100, 1000, and 10,000 iterations are shown in (b), (c), and (d), respectively.

Simulation with quantum-photon noise at an average of 4000 photons per cell. (a) Original nondegraded signal. (b) Degraded signal. (c) Initial guess for the first iteration. (d) Restoration at 1000 iterations.

Two-dimensional simulation with quantum-photon noise: (a) original nondegraded signal; (b) diffraction-limited, quantum-limited image at an average of 160 photons per pixel over a 64 × 64 image; (c) truncated version of (b); (d) restoration after 50,000 iterations.